How do I get the object to follow the Mixamo Rig
There is an Armature of Trooper and Armature of Lightsaber consist from one bone.


Parent Lightsaber to this bone - select Lightsaber add Armature modifier > Object = Armature.001 and Vertex Group > Bone 





Parent this bone to Troopers hand bone - in this case seems to me enough to select bone in pose mode add Bone Constraint > Copy Location > Target Armature > Bone mixamorig.RightHand and enable Offset to adjust proper position in hand.




On frame 60 the left hand if shifting so you would have to compensate that by Lightsaber rotation in that direction. Since Rotation Constraint doesnt use offset (like Loc does). You will need 


add Empty
Parent to bone "LeftHandMiddle1"
for Lightsaber add Bone Constraint > Track to Empty
position Empty somewhere in a middle of fist




For animating object holded by two hands would be better snap hands and animate object. Like here Rigging two handed weapon. Animation from mixamo has quite a lot shifted hands so they dont match for holding stick (like lightsaber), fingers are also out of healthy human position :) nothing for camera close look.